To stabilize your mixing bowl, position a wet dishcloth twisted into a ring on the countertop. Place a large mixing bowl on the dishcloth ring and vigorously whisk together the beaten egg yolks, mustard, lemon juice and vinegar for about 30 seconds to 1 minute.

Pour oils into a 2-cup measuring cup with a spout. Whisking constantly and briskly, add oil in a slow steady stream. (Note: Finished mayonnaise should have a smooth, well-blended consistency. If it appears to have separated, continue whisking vigorously to emulsify.) Season to taste with salt and pepper.

Make It Ahead: Make recipe and keep refrigerated in an airtight container for up to 2 days.